\documentclass[twoside]{article} \pagestyle{myheadings} \markboth{\hfil A non-local problem with integral conditions \hfil EJDE--1999/45} {EJDE--1999/45\hfil L. S. Pulkina \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 1999}(1999), No. 45, pp. 1--6. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A non-local problem with integral conditions for hyperbolic equations \thanks{ {\em 1991 Mathematics Subject Classifications:} 35L99, 35D05. \hfil\break\indent {\em Key words and phrases:} Non-local problem, generalized solution. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted July 29, 1999. Published November 15, 1999.} } \date{} % \author{L. S. Pulkina} \maketitle \begin{abstract} A linear second-order hyperbolic equation with forcing and integral constraints on the solution is converted to a non-local hyperbolic problem. Using the Riesz representation theorem and the Schauder fixed point theorem, we prove the existence and uniqueness of a generalized solution. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \section{Introduction} Certain problems arising in: plasma physics [1], heat conduction [2, 3], dynamics of ground waters [4, 5], thermo-elasticity [6], can be reduced to the non-local problems with integral conditions. The above-mentioned papers consider problems with parabolic equations. However, some problems concerning the dynamics of ground waters are described in terms of hyperbolic equations [4]. Motivated by this, we study the equation $$\label{eq1} Lu\equiv u_{xy} + A(x,y)u_x + B(x,y)u_y + C(x,y)u = f(x,y)$$ with smooth coefficients in the rectangular domain $$D=\{ (x,y): 0 {1\over c_1}, and let$$ S_\lambda (w)=S(\sqrt 2 \lambda w)\,.$$\begin{theorem} If  \bar f(x,y) \in L_2(D) and |\bar f(x,y)|\leq {P\over \sqrt 2}, then there exists at least one generalized solution  w_0 \in \tilde {H}^1(D) to problem~(\ref {eq5})-(\ref{eq6}), where \|w_0\|^2_1 \leq {P^2 \over \eta ^2}, with  \eta ^2 = c_1^2 - {1\over \lambda ^2}. Furthermore, the solution is uniquely determined, if c_2From (\ref{eq7}) and Lemma 1 we have$$ \|w_1-w_2\|_1 \leq {1\over c_1}\|Sw_1-Sw_2\|_{L_2} \leq {c_2\over c_1}\|w_1-w_2\|_1.  Thus, if \$ c_2